In [1]}, sampling \(\mathbf {x}\sim p_{\mathbf {x}}(\mathbf {x})\)  is straightforward since they focus on autoregressive language models.
In the context of latent-variable vision generative models, where \(\mathbf {x}\sim p_{\mathbf {x}}(\mathbf {x})\)  is implicitly defined as \(\mathbf {z}\sim p_{\mathbf {z}}(\mathbf {z}), \mathbf {x}= G(\mathbf {z})\),
Eq. (REF ) is equivalent to
\(\hat{\mathbf {\beta }} = \mathop {\arg \min }_{\mathbf {\beta }} \mathbb {E}_{\mathbf {z}^{(1)}, \ldots , \mathbf {z}^{(N)} \overset{\text{i.i.d.}}{\sim }p_{\mathbf {z}}(\mathbf {z}), \mathbf {x}^{(j)} = G(\mathbf {z}^{(j)}) } \bigg \Vert \frac{\sum _{j=1}^{N}\exp \big (\mathbf {\beta }^\top \mathbf {\gamma }(\mathbf {x}^{(j)})\big )\mathbf {\gamma }(\mathbf {x}^{(j)})}{\sum _{j^{\prime }=1}^{N}\exp \big (\mathbf {\beta }^\top \mathbf {\gamma }(\mathbf {x}^{(j^{\prime })})\big )} - \mathbf {\mu }\bigg \Vert _2^2.\) 
